Challenges to Metaphysical Realism

According to metaphysical realism, the world is as it is independently
of how humans or other inquiring agents take it to be. The objects the
world contains, together with their properties and the relations they
enter into, fix the world’s nature and these objects exist
independently of our ability to discover they do. Unless this is so,
metaphysical realists argue, none of our beliefs about our world could
be objectively true since true beliefs tell us how things are and
beliefs are objective when true or false independently of what anyone
might think.

Many philosophers believe metaphysical realism is just plain common
sense. Others believe it to be a direct implication of modern science,
which paints humans as fallible creatures adrift in an inhospitable
world not of their making. Nonetheless, metaphysical realism is
controversial. Besides the analytic question of what it means to
assert that objects exist independently of the mind, metaphysical
realism also raises epistemological problems: how can we obtain
knowledge of a mind-independent world? There are also prior semantic
problems, such as how links are set up between our beliefs and the
mind-independent states of affairs they allegedly represent. This is
the Representation Problem.

Anti-realists deny the world is mind-independent. Believing the
epistemological and semantic problems to be insoluble, they conclude
realism must be false. In this entry I review a number of semantic and
epistemological challenges to realism all based on the Representation
Problem:

The Manifestation Argument: the cognitive and linguistic behaviour
of an agent provides no evidence that realist mind/world links
exist;

The Language Acquisition Argument: if such links were to exist
language learning would be impossible;

The Brain-in-a-Vat Argument: realism entails both that we could be
massively deluded (‘brains in a vat’) and that if we were
we could not even form the belief that we were;

The Conceptual Relativity Argument: it is senseless to ask what
the world contains independently of how we conceive of it, since the
objects that exist depend on the conceptual scheme used to classify
them;

The Model-Theoretic Argument: realists must either hold that an
ideal theory passing every conceivable test could be false or that
perfectly determinate terms like ‘cat’ are massively
indeterminate, and both alternatives are absurd.

I proceed by first defining metaphysical realism, illustrating its
distinctive mind-independence claim with some examples and
distinguishing it from other doctrines with which it is often
confused, in particular factualism. I then outline the Representation
Problem in the course of presenting the anti-realist challenges to
metaphysical realism that are based on it. I discuss metaphysical
realist responses to these challenges, indicating how the debates have
proceeded, suggesting various alternatives and countenancing
anti-realist replies.

The aim throughout will be to see whether the realist can respond to
the anti-realist challenges and so much of the subsequent discussion
will be taken up with attempts to formulate realist replies to these
challenges that have some initial credibility. However it needs to be
noted that all these replies are provisional: anti-realism gains much
of its force by highlighting a gap between realist metaphysics and
epistemology that no one really knows how to bridge.

Metaphysical realism is the thesis that the objects, properties and
relations the world contains exist independently of our thoughts about
them or our perceptions of them. Anti-realists either doubt or deny
the existence of the entities the metaphysical realist believes in or
else doubt or deny their independence from our conceptions of
them.

Metaphysical realism is not the same as scientific realism. That the
world’s constituents exist mind-independently does not entail
that its constituents are as science portrays them. One could adopt an
instrumentalist attitude toward the theoretical entities posited by
science, continuing to believe that whatever entities the world
actually does contain exist independently of our conceptions and
perceptions of them.

Henceforth, we shall often just use the term ‘realism’ to
mean metaphysical realism. Opposition to realism can take many forms
so there is no single theoretical view denoted by the term
‘anti-realism’. One approach, popular in continental
philosophy, is to reject realism in favour of the view that words can
only acquire their meaning intra-linguistically, through
their semantic relations with other words, rather than through any
(fanciful) ‘referential’ relations to the world outside of
language.

Within the ranks of analytic philosophy, verificationists and
pragmatists also reject realism, though for different reasons. We
shall focus in this entry on the types of criticism voiced by these
two groups of analytic philosophers with Michael Dummett advocating
verificationism and Hilary Putnam pragmatism. Both reject realism by
deploying semantic considerations in arguments designed to show that
realism is untenable. The goal of this entry is to outline these
‘semantic’ challenges to realism and to see whether they
can be answered.

This characterization of realism in terms of mind-independence is not
universally accepted. Some object that mind-independence is
obscure. Others maintain that realism is committed, in addition, to a
distinctive (and tendentious) conception of truth [Putnam 1981, 1985,
1992; Wright 1991] or, more radically, that realism just is a thesis
about the nature of truth—that truth can transcend the
possibility of verification, ruling statements for which we can gather
no evidence one way or the other to be determinately either true or
false. An example would be “Julius Caesar’s heart skipped
a beat as he crossed the Rubicon.” Thus the realist on this view
is one who believes the law of bivalence (every statement is either
true or false) holds for all meaningful (non-vague) statements
[Dummett 1978, 1991, 1993].

These semantic formulations of metaphysical realism are unacceptable
to realists who are deflationists about truth, denying that truth is a
substantive notion which can be used to characterise alternative
metaphysical views [see the
entry on the deflationary theory
of truth]. Such realists tend to ignore the anti-realist’s
semantic and epistemological challenges to their position.

It is a mistake to identify realism with factualism, the view that
sentences in some discourse or theory are to be construed literally as
fact-stating ones. The anti-realist views discussed below are
factualist about discourse describing certain contentious
domains. Adopting a non-factualist or error-theoretic interpretation
of some domain of discourse commits one to anti-realism about its
entities. Factualism is thus a necessary condition for realism. But it
is not sufficient. Verificationists like Dummett reject the idea that
something might exist without our being able to recognize its
existence. They can be factualists about entities such as numbers and
quarks while maintaining anti-realism about them since they deny that
any entities can exist mind-independently.

Why do some find the notion of mind-independent existence inadequate
for the task of formulating metaphysical realism? The most common
complaint is that the notion is either obscure, or, more strongly,
incoherent or cognitively meaningless. An eloquent spokesman for this
strong view was Rudolf Carnap: “My friends and I have maintained
the following theses,” Carnap announces [Carnap 1963,
p.868]:

(1) The statement asserting the reality of the external world
(realism) as well as its negation in various forms, e.g. solipsism and
several forms of idealism, in the traditional controversy
are pseudo-statements, i.e., devoid of cognitive content. (2)
The same holds for the statements about the reality or irreality
of other minds (3) and for the statements of the reality or
irreality of abstract entities (realism of universals or
Platonism, vs. nominalism).

In spite of his finding these disputes meaningless, Carnap indicates
how he thinks we could reconstruct them (sic.) so as to make some
sense of them: if we were to “replace the ontological theses
about the reality or irreality of certain entities, theses which we
regard as pseudo-theses, by proposals or decisions concerning the use
of certain languages. Thus realism is replaced by the practical
decision to use the reistic language”.

Carnap does not have in mind a factualist reformulation of
metaphysical realism here—his “reistic” language is
strictly limited to the description of “intersubjectively
observable, spatio-temporally localized things or events”.

What matters for our purposes is not Carnap’s sense of a
commensurability between a metaphysical thesis about reality and a
practical decision to speak only about observable things, but rather
that he thinks he can explain how the illusion of meaningfulness
arises for the metaphysical theses he declares “devoid of
cognitive content”.

His explanation has to do with a distinction between two types of
questions: internal and external questions. By way
of illustration Carnap shows how the distinction works in the
controversy over the existence of abstract entities:

An existential statement which asserts that there are entities of a
specified kind can be formulated as a simple existential statement in
a language containing variables for these entities. I have called
existential statements of this kind, formulated within a
given language, internal existential statements. [Carnap
1963, p. 871]

Carnap contends that

Just because internal statements are usually analytic and trivial, we
may presume that the theses involved in the traditional philosophical
controversies are not meant as internal statements, but rather
as external existential statements; they purport to assert
the existence of entities of the kind in question not merely within a
given language, but, so to speak, before a language has been
constructed. [1963, p. 871]

Declaring all such external existential questions devoid of cognitive
content, Carnap now feels emboldened to dismiss both realism that
asserts the ontological reality of abstract entities and nominalism
that asserts their irreality as “pseudo-statements if they claim
to be theoretical statements” (ibid).

More importantly, Carnap has hit upon an explanation for the
persistent allure of the notion of mind-independent reality: we often
wish to know whether some existence claim is true. Provided we realize
existence claims can only be properly formulated and evaluated within
a language \(L\) our query is perfectly reasonable and can very
often be answered by examining the specification of the domain
of \(L\)’s quantifiers. Thus, to use Carnap’s own
example, suppose a theorist wishes to know for a
language \(L'\) whose domain contains material objects,
classes of objects and classes of classes of objects, the answer to
the following existential question:

(1)
Does there exist an \(x\) and a \(y\) such
that \(x\) is an element of an element of \(y\)?

Carnap maintains (ibid) that an affirmative answer to this
question is both true and provable in \(L'\) (though he
does not specify any theory expressible in \(L'\) in which
(1) is derivable).

But suppose that instead of \(L'\), our theorist had asked
the same question of another language \(L''\) the universe
of discourse for which contained material objects and classes of these
but no classes of classes of them. Then the following universal
statement would now be provable in \(L''\), Carnap claims
(again without specifying any theory expressible in \(L''\)
for which this might hold), as well as true in that language:

(2)
For every \(x\) and \(y\), \(x\) is not an element
of an element of \(y\)

Suppose our theorist, Al let’s call him, though initially
attracted to \(L'\) for its superior expressive and
deductive power when compared to \(L''\), now starts to have
misgivings about the content and consistency of some of its
existential assertions. After deliberating he decides:

(3)
There are classes of objects.

But he also accepts:

(4)
There are no classes of classes of objects.

This brings him into conflict with his good friend Bob. For Bob
believes not only in classes of objects but also in classes of classes
of objects and thus endorses \(L'\) as a language best
suited to represent his ontological beliefs. That is, like Al, Bob
believes (3), but Bob also accepts (5):

(5)
There are classes of classes of objects.

Is there a genuine dispute between Al and Bob? Is there a fact of the
matter as to who is right, whose ontological views reflect the way the
world is really structured? Carnap says “No”: seeking to
elevate the modal status of their linguistic decisions from mere
preferences for one language over another to unconditional obligations
to reflect how reality is independently of any representaton,
Al and Bob have temporarily lost sight of the particular linguistic
contexts that give meaning to the existential claims they respectively
advanced at (4) and (5). All that can be meaningfully said, according
to Carnap, is that whilst (4) is true in \(L''\) it is false
in \(L'\) and, conversely, whilst (5) is true
in \(L'\), it is false in \(L''\). The cognitive
content of (5) for Bob is given by (1) and that of (4) for Al by
(2). As Carnap puts it:

Thus we see the difference between (them) is not a difference
in theoretical beliefs [DK: as Bob seems to think when he
makes the pseudo-assertion at (5)]; it is merely
a practical difference in preferences and decisions
concerning the acceptance of languages. [loc. cit., p. 873]

This is a beguiling story but it does not do what Carnap wishes it to
do: it does not spirit away all troubling metaphysical questions about
mind-independent existence by parlaying them into (or replacing them
with) sanitized questions about the entities the quantifiers range
over in this or that language. Here’s why. Consider the
following case. Suppose the year is 1928, the year Carnap published
his Aufbau. A mathematician, Cass, working in classical
mathematics (sometimes shortened to “CM”), comes across
the following question:

(Q)
Are there irrational numbers \(a\) and \(b\) such
that \(a^b\) is rational?

Cass realizes at once that she can answer this question, reasoning
from premise (A):

(A)
Either \(\sqrt{2}^{\sqrt{2}}\) is rational or it is
irrational.

The reasoning continues:

Suppose this number \(\sqrt{2}^{\sqrt{2}}\) is rational. Then since
\(\sqrt{2}\) is irrational, our problem is solved by taking \(a =
\sqrt{2}, b = \sqrt{2}\). Suppose alternatively that
\(\sqrt{2}^{\sqrt{2}}\) is irrational. Then that very irrational
number raised to the power \(\sqrt{2}\) must be rational. For this
number is equivalent to
\(\sqrt{2}^{\sqrt{2} \times \sqrt{2}}\)
which is just the rational number 2. So in this latter case, by
selecting \(a = \sqrt{2}^{\sqrt{2}}, b = \sqrt{2}\) we ensure that
we have selected two irrational numbers \(a\), \(b\) such that \(a^b\) is
rational.

Now as the background logic for CM is classical there is nothing wrong
with Cass’s reasoning proceeding, as it does, from an instance
of the classically valid Law of Excluded Middle at (A). It is an
example of a “non-constructive” existence proof:
demonstrating that one or another alternative must hold without
providing a means for ascertaining which one does hold.

Suppose now we ask Cass which of the two statements below is true in
classical mathematics:

\(\sqrt{2}\) is an irrational number such that
\(\sqrt{2}^{\sqrt{2}}\) is a rational number.

\(\sqrt{2}^{\sqrt{2}}\) and \(\sqrt{2}\) are irrational numbers
such that \((\sqrt{2}^{\sqrt{2}})^{\sqrt{2}}\)
is a rational number.

Cass, working in 1928, believes one or the other of these statements
must be true in classical mathematics but she has no means for
determining which is true. So she cannot answer our question. Further,
let us suppose that no one ever does find a method for
determining which alternative holds good.

Aside: As it turns out (and this is the reason for indexing
the example to a particular time) this last supposition is contrary to
fact. In 1934 Gelfond and Schneider independently proved that
if \(a, b\) are algebraic numbers with \(a \ne 0\) or \(1\)
and \(b\) not rational then any value
of \(a^b\) [\(= \exp(b \log a)\)]
is a transcendental number. The Gelfond-Schneider Theorem answered in
the affirmative David Hilbert’s Seventh Problem: whether
\(2^{\sqrt{2}}\) is transcendental. It also follows that
\(\sqrt{2}^{\sqrt{2}}\) is irrational so that (II) is
true-in-CM.

Now even though she lacks any method for deciding which alternative
holds, according to Cass either (i) is true-in-CM or else (ii) is
true-in-CM. But if so, Cass in 1928 has an instance of a
mind-independent existence claim holding of an internal
existence statement: one of these two pairs of numbers
\((a, b)\) in question, either
\((a=\sqrt{2}, b=\sqrt{2})\) or else
\((a=\sqrt{2}^{\sqrt{2}}, b=\sqrt{2})\) is such
that \(a\) and \(b\) are both irrational
but \(a^b\) is rational even though we may
never be able to determine which pair of numbers it is (and
we could have been in Cass’s situation today had the
Gelfond-Schneider Theorem lain forever undiscovered).

But what should Carnap say about this case? He cannot protest that
Cass’s assertion that one or other element of the
sentence pair {i), (ii)} is true-in-CM is a pseudo-statement as he did
for Bob’s assertion (5) that there are classes of classes of
objects. For the statement that either (i) is true-in-CM or else (ii)
is true-in-CM is an internal statement and thus, by
Carnap’s lights, a statement that has cognitive content.

Hence, one cannot undermine the notion of mind-independent reality in
the simple way Carnap imagines, namely, by the internal/external
distinction coupled with the claim that external statements are
pseudo-statements. For, whatever its other virtues, the
internal/external distinction cannot explain why someone should
believe that exactly one of (i) or (ii) has to be true in a
certain language even though we might never be in a position to
determine which. And this is precisely what the belief in
mind-independent reality amounts to.

The first anti-realist challenge to consider focuses on the use we
make of our words and sentences. The challenge is simply this: what
aspect of our linguistic use could provide the necessary evidence for
the realist’s correlation between sentences and mind-independent
states of affairs? Which aspects of our semantic behaviour manifest
our grasp of these correlations, assuming they do hold?

For your representations of the world to be reliable, there must be a
correlation between these representations and the states of affairs
they portray. So the cosmologist who utters the statement “the
entropy of the Big Bang was remarkably low” has uttered a truth
if and only if the entropy of the Big Bang was remarkably low.

A natural question to ask is how the correlation between the statement
and the mind-independent state of affairs which makes it true is
supposed to be set up. One suggestive answer is that the link is
effected by the use speakers make of their words, the statements they
endorse and the statements they dissent from, the rationalizations
they provide for their actions and so forth; cognitively, it will be
the functional role of mental symbols in thought, perception and
language learning etc. that effects these links.

When we look at how speakers actually do use their sentences,
anti-realists claim, we see them responding not to states of affairs
that they cannot in general detect but rather to agreed upon
conditions for asserting these sentences. Scientists assert “the
entropy of the Big Bang was remarkably low” because they all
concur that the conditions justifying this assertion have been
met.

What prompts us to use our sentences in the way that we do are the
public justification conditions associated with those sentences,
justification conditions forged in linguistic practices which imbue
these sentences with meaning.

The realist believes we are able to mentally represent
mind-independent states of affairs. But what of cases where everything
that we know about the world leaves it unsettled whether the relevant
state of affairs obtains? Did Socrates sneeze in his sleep the night
before he took the hemlock or did he not? How could we possibly find
out? Yet realists hold that the sentence “Socrates sneezed in
his sleep the night before he took the hemlock” will be true if
Socrates did sneeze then and false if he did not and that this is a
significant semantic fact.

The Manifestation challenge to realism is to isolate some feature of
the use agents make of their words, or their mental symbols, which
forges the link between mind-independent states of affairs and the
thoughts and sentences that represent them. Nothing in the
thinker’s linguistic behaviour, according to the anti-realist,
provides evidence that this link has been forged—linguistic use
is keyed to public assertibility conditions, not undetectable
truth-conditions. In those cases, such as the Socrates one, where we
cannot find out whether the truth-condition is satisfied or not, it is
simply gratuitous to believe that there is anything we can think or
say or do which could provide evidence that the link has been set up
in the first place. So the anti-realist claims [Dummett 1978, 1991,
1993 Tennant 1997; Wright 1993].

Why should we expect the evidence to be behavioural rather than, say,
neurophysiological? The reason anti-realists give is that the meanings
of our words and (derivatively for them) the contents of our thoughts
are essentially communicable and thus must be open for all speakers
and thinkers to see [Dummett 1978, 1993].

The second challenge to be considered concerns our acquisition of
language. The challenge to realism is to explain how a child could
come to know the meanings of certain sentences within his/her
language: the ones which the realist contends have undetectable
truth-makers associated with them. How could the child learn the
meanings of such sentences if these meanings are determined by states
of affairs not even competent speakers can detect?

Consider the sentence (S) once more:

(S)
Socrates sneezed in his sleep the night before he took the
hemlock.

Realists say (S) is either true or false even though we may (and
almost certainly will) never know which it is. The state of affairs
which satisfies (S)’s truth-condition when it is true, its
‘truthmaker’, and the state of affairs which satisfies the
truth-condition of the negation of (S) when (S) is false are supposed
to be able to hold even though competent speakers cannot detect
whether they do. How could the child ever learn about this
undetectable relation?

Suppose God (or nature) had linked our mental representations to just
the right states of affairs in the way required by the realist. If so,
this is a semantically significant fact. Anyone learning their native
language would have to grasp these correspondences between sentences
and states of affairs. How can they do this if even the competent
speakers whom they seek to emulate cannot detect when these
correspondences hold? In short, competence in one’s language
would be impossible to acquire if realism were true [Dummett 1978,
1993; Wright 1993]. This is the Language Acquisition challenge.

This challenge is exacerbated by the anti-realist’s assumption
that since the linguistic meaning of an expression \(E\) is
determined solely by competent speakers’ use of \(E\) the
child’s task in all cases is to infer the meaning of \(E\)
from its use. Thus Dummett [1978 pp. 216–217], in discussing the
meaning of mathematical statements, proposes a thesis he argues holds
for the meanings of every kind of statement:

The meaning of a mathematical statement determines and is exhaustively
determined by its use. The meaning of a mathematical statement cannot
be, or contain as an ingredient, anything which is not manifest in the
use made of it, lying solely in the mind of the individual who
apprehends that meaning: if two individual agree completely about the
use to be made of the statement, then they agree about its
meaning. The reason is that the meaning of a statement consists solely
in its role as an instrument of communication between individuals,
just as the powers of a chess-piece consist solely in its role in the
game according to the rules.

W.V.O. Quine is even more insistent on the public nature of linguistic
meaning. Displaying his unshakable faith in Skinnerian models of
language-learning he writes [1992, pp. 37–38]:

In psychology one may or may not be a behaviourist, but in linguistics
one has no choice … There is nothing in linguistic meaning
beyond what is to be gleaned from overt behaviour in observable
circumstances.

According to Hilary Putnam, the metaphysical realist subscribes not
just to the belief in a mind-independent world but also to the thesis
that truth consists in a correspondence relation between words (or
mental symbols) and things in that mind-independent world. Call this
thesis correspondence truth (after Devitt 1991). More
importantly, metaphysical realists aver that an ideal theory of the
world could be radically false, Putnam contends:
‘radical’ in the sense that all (or almost all) of
the theory’s theses could fail to hold. Such a global failure
would result if we were to be ‘brains-in-a-vat’ our brains
manipulated by mad scientists (or machines, as in the movie The
Matrix) so as to dream of an external world that we mistake for
reality. Call this thesis radical skepticism.

It is widely believed that states of affairs that are truly
mind-independent do engender radical skepticism. The skeptic contends
that for all we could tell we could be brains in a vat—brains
kept alive in a bath of nutrients by mad alien scientists. All our
thoughts, all our experience, all that passed for science would be
systematically mistaken if we were. We’d have no bodies although
we thought we did, the world would contain no physical objects, yet it
would seem to us that it did, there’d be no Earth, no Sun, no
vast universe, only the brain’s deluded representations of
such. At least this could be the case if our representations derived
even part of their content from links with mind-independent objects
and states of affairs. Since realism implies that such an absurd
possibility could hold without our being able to detect it, it has to
be rejected, according to anti-realists.

A much stronger anti-realist argument due to Putnam uses the
brain-in-a-vat hypothesis to show that realism is internally
incoherent rather than, as before, simply false. A crucial assumption
of the argument is semantic externalism, the thesis that the reference
of our words and mental symbols is partially determined by contingent
relations between thinkers and the world. This is a semantic
assumption many realists independently endorse.

Given semantic externalism, the argument proceeds by claiming that if
we were brains in a vat we could not possibly have the thought that we
were. For, if we were so envatted, we could not possibly mean by
‘brain’ and ‘vat’ what unenvatted folk mean by
these words since our words would be connected only to neural impulses
or images in our brains where the unenvatteds’ words are
connected to real-life brains and real-life vats. Similarly, the
thought we pondered whenever we posed the question “am I a brain
in a vat?” could not possibly be the thought unenvatted folk
pose when they ask themselves the same-sounding question in
English. But realism entails that we could indeed be brains in a
vat. As we have just shown that were we to be so, we could not even
entertain this as a possibility, realism is incoherent [Putnam
1981].

If the notion of mind-independent existence is incoherent, as
anti-realists contend, what should we put in its stead? Berkeley
famously answered “Mind-dependent existence!” where the
Mind in question, for the good Bishop, was, of course, the Mind of
God. Modern anti-realists tend not to be theists and tend not to
relativize existence to any single mind. Instead of God they posit
conceptual schemes as that on which the notion of existence
depends. To that extent they follow Kant rather than Berkeley, though
unlike Kant they tend to be pluralists—it is conceptual schemes
which they endorse rather than a single transcendental scheme which
Kant held to be obligatory for all rational creatures.

According to this view, there can no more be an answer to the question
“What objects and properties does the world contain?”
outside of some scheme for classifying entities than there can be an
answer to the question of whether two events \(A\) and \(B\)
are simultaneous outside of some inertial frame for dating those
events. The objects which exist are the objects some conceptual scheme
says exists—‘mesons exist’ really means
‘mesons exist relative to the conceptual scheme of current
physics’.

Realists think there is a unitary sense of ‘object’,
‘property’ etc., for which the question “what
objects and properties does the world contain?” makes sense. Any
answer which succeeded in listing all the objects, properties, events
etc. which the world contains would comprise a privileged description
of that totality. Anti-realists reject this. For them
‘object’, ‘property’ etc., shift their senses
as we move from one conceptual scheme to another. Some anti-realists
argue that there cannot be a totality of all the objects the world
contains since the notion of ‘object’ is indefinitely
extensible and so, trivially, there cannot be a privileged description
of any such totality.

How does the anti-realist defend conceptual relativity? One way is by
arguing that there can be two complete theories of the world which are
descriptively equivalent yet logically incompatible from the
realist’s point of view. For example, theories of space-time can
be formulated in one of two mathematically equivalent ways: as an
ontology of points, with spatiotemporal regions being defined as sets
of points; or as an ontology of regions, with points being defined as
convergent sets of regions. Such theories are descriptively equivalent
since mathematically equivalent and yet are logically incompatible
from the realist’s point of view, anti-realists contend [Putnam
1985, 1990].

Putnam’s Model-Theoretic Argument is the most technical of the
arguments we have so far considered although we shall not reproduce
all the technicalities here. The central ideas can be conveyed
informally, although some technical concepts will be mentioned where
necessary. The argument purports to show that the Representation
Problem—to explain how our mental symbols and words get hooked
up to mind-independent objects and how our sentences and thoughts
target mind-independent states of affairs—is insoluble.

According to the Model-Theoretic Argument, there are simply too many
ways in which our mental symbols can be mapped onto items in the
world. The consequence of this is a dilemma for the realist. The first
horn of the dilemma is that s/he must accept that what our symbols
refer to is massively indeterminate. The second horn is that s/he must
insist that even an ideal theory, whose terms and predicates can
demonstrably be mapped veridically onto objects and properties in the
world might still be false, i.e., that such a mapping might not be the
right one, the one ‘intended’.

Neither alternative can be defended, according to
anti-realists. Concerning the first alternative, massive indeterminacy
for perfectly determinate terms is absurd. As for the second, for
realists to contend that even an ideal theory could be false is to
resort to unmotivated dogmatism, since on their own admission we
cannot tell which mapping the world has set up for us. Such dogmatism
leaves the realist with no answer to a skepticism which undermines any
capacity to reliably represent the world, anti-realists maintain.

It might be useful to informally illustrate some basic ideas
underlying the Model-Theoretic Argument. The discussion to follow
trades formal precision for intuitive accessibility. In logic, a
theory is a set of sentences. We are going to consider what happens to
the interpretation of 3 simple sentences that comprise our theory when
we vary the way we refer to the individuals those sentences talk about
and also vary how we classify those individuals. So, imagine that you
and your four year old niece Maddy are fortunate enough to be watching
three elite sprinters at a training session—Usain Bolt, Justin
Gatlin and Assafa Powell. Suppose we let the letter b stand for Bolt,
g stand for Gatlin and p stand for Powell. These letters are called
individual constants. We will also need a single predicate letter J
representing the English predicate ‘is Jamaican’ to
formulate our 3 sentence theory. In logic we distinguish between
theories (sets of sentences) and the things those theories talk about
(usually not sentences). The collection of items the theory talks
about feature in abstract entities known as structures as the domain
of the structure and what the theory says about those items receives
an interpretation in the structure. An interpretation function assigns
objects from the domain of a structure to individual constants such as
our b, g and p and sets of objects (subsets of the domain) to monadic
predicates such as our J. If we wish to express relations between
objects such as one individual being faster than or taller than
another (binary relations) or one individual standing between two
others (ternary relations) we will need sets of ordered pairs (for the
binary case) or ordered triples (for the ternary case) from the
domain. Generally, n-place relations will require our interpretation
function to supply n-tuples of objects from the domain as extensions
for n-place predicates. By ‘extension’ of a predicate we
simply mean the (sets of) things the predicate applies to.

We are going to focus on the simplest case. Our theory asserts that:
(1) Bolt is Jamaican, (2) Powell is Jamaican, (3) Gatlin is not
Jamaican. We shall write these sentences as (1) \(Jb\), (2) \(Jp\),
(3) \(\neg Jg\) respectively. This simple theory is true. When you
inform Maddy of these facts you mean to refer to the man Usain Bolt
when you use the name ‘Bolt’, to Assafa Powell by the name
‘Powell’ and to Justin Gatlin by ‘Gatlin’. You
also mean to be describing the first two sprinters as Jamaican and the
last as non-Jamaican when you use the predicate ‘is
Jamaican’ in sentences (1), (2) and (3).

A structure in which \(b\) denotes Bolt, \(p\) denotes Powell and \(g\) denotes
Gatlin and in which the extension of \(J\) is {Bolt, Powell}
will make the sentences (1), (2) and (3) all true. Such a structure is
said to be a model of the theory. Furthermore, as the model represents
the names ‘Bolt’, ‘Powell’,
‘Gatlin’ as applying to exactly the right individuals and
the predicate ‘is Jamaican’ as applying to just the right
set of individuals we say the model is an intended
model.

However, it is perfectly possible for a structure for a theory to make
all its sentences true (and thus be a model of the theory) without
that structure being an intended model. Suppose Maddy
mistakes which particular individuals you are referring to when you
use the names ‘Bolt’, ‘Powell’ and
‘Gatlin’. Perhaps these sprinters lined up in that order
for one trial and now line up in a different order for the next
trial. Suppose she is also misinformed about what ‘is
Jamaican’ describes. Maybe she thinks it applies to a sprinter
in the outside or middle lane and although that was the order for Bolt
and Powell in the first trial, in the second one which is just about
to start Gatlin and Bolt are in the outside and middle lanes.

Let us use \(M\) to denote the intended model of the 3 sentence theory
and use \(M^*\) to denote Maddy’s non-standard model. Suppose we
use the following notation to mean the individual constant \(b\)
refers to Bolt in the model \(M\): \(|b|_M =\) Bolt and so we also
represent the facts that in \(M\) the constant g refers to Gatlin and
\(p\) refers to Powell respectively as \(|g|_M =\) Gatlin and \(|p|_M =\)
Powell. Suppose finally that we symbolize the interpretation of the
predicate \(J\) in \(M\) by \(|J|_M =\) {Bolt, Powell}.

We can now compute the truth-values of sentences such as \(Jb\),
\(Jg\) and \(Jp\) in the model \(M\). Simple sentences such as these will
turn out true in the model if the individual denoted by the individual
constant in the model is included in the set of individuals comprising
the extension of the predicate \(J\) in the model. Since the individuals
assigned to b and p are indeed included in the set that comprises the
extension of \(J\) in \(M\), viz. { Bolt, Powell }, we have \(Jb\) and
\(Jp\) coming out true in \(M\). However since Gatlin is not included in
the set { Bolt, Powell } \(Jg\) comes out false in \(M\), whence \(\neg Jg\)
comes out true in \(M\). This is exactly as it should be: Usain Bolt and
Assafa Powell are both Jamaican sprinters but Justin Gatlin is an
American rather than Jamaican sprinter. We represent these truths as:
(i) \(|Jb|_M =\) True, (ii) \(|Jg|_M =\) False, (iii) \(|Jp|_M =\)
True.

Maddy’s model \(M*\) is a non-standard or unintended one. She
thinks the name ‘Bolt’ refers to Gatlin, the name
‘Powell’ refers to Bolt and the name
‘Gatlin’refers to Powell. Furthermore she thinks the
predicate ‘is Jamaican’ applies to Gatlin and Bolt but not
to Powell. That is, we have that in \(M*\): \(|b|_{M*} =\) Gatlin,
\(|g|_{M*} =\) Powell, \(|p|_{M*} =\) Bolt and for \(J\) our sole
predicate \(|J|_{M*} =\) { Gatlin, Bolt }.

In spite of her misunderstanding the intended referents for the
sprinters’ names and the intended extension of the predicate
‘is Jamaican’, Maddy’s model \(M*\) assigns exactly
the right truth-values for the 3 sentences above as readers can check
for themselves. That is: (i) \(|Jb|_{M*} =\) True, (ii) \(|Jg|_{M*}
=\) False, (iii) \(|Jp|_{M*} =\) True. The model \(M*\) is said to be
a permuted model of \(M\). It is as if the objects in the
domain were systematically shuffled around whilst the labels were kept
fixed, as Tim Button puts it [Button (2013). The example in the text
here was inspired by Button’s Fig 2.1 p.15].

Now consider a very different structure \(N\) the domain of which
consists only of the natural numbers 3, 4 and 5. We are going to
interpret our simple sprinter theory in \(N\). So suppose \(|b|_N =
3\), \(|g|_N = 4\), \(|p|_N = 5\). Suppose also that our predicate \(J\)
is interpreted in \(N\) so as to have the same extension in \(N\) as
the English predicate ‘is an odd number’, i.e. \(|J|_N =\)
{3,5}. Then, clearly \(N\) is a model of our three sentence theory
since we have (i) \(|Jb|_N =\) True, (ii) \(|Jg|_N =\) False, (iii)
\(|Jp|_N =\) True. Let us call structures whose domains consist of
numbers ‘numeric’ structures.

The nub of Putnam’s Model-Theoretic Argument against realism is that the realist cannot distinguish the intended model for his/her total theory of the world from non-standard interlopers such as permuted models or ones derived from numeric models, even when total theory is a rationally optimal one that consists, as it must do, of an infinite set of sentences and the realist is permitted to impose the most exacting constraints to distinguish between models. This is a very surprising result if true! How does Putnam arrive at it?

Putnam actually uses a number of different arguments to establish the conclusion above. The one of most concern to realists, as Taylor (2006) emphasises, is the one based on Gödel‘s Completeness Theorem. For this argument purports to prove that an ideal theory of the world could not be false, a conclusion flatly inconsistent with realism. It will be useful to first state the logical theorems on which the argument is based.

Let us start with the Completeness Theorem. In 1930 Kurt Gödel proved that a certain type of predicate logic, first-order logic without identity (which we shall sometimes denote as FOL), is complete in the sense that all sentences of that logic that are true under every interpretation can be derived within that logic. This means that every set of FOL sentences S that is ‘syntactically’ consistent (i.e. consistent in the sense that no contradiction can be derived from S within this logic), also has a model [See the entry on Kurt Gödel for further details and a proof of the theorem].

The other theorem we shall need for the Model-Theoretic Argument below goes by the name of the Löwenheim-Skolem Theorem. To understand this theorem, one needs to first know something of the work of the nineteenth century mathematician Georg Cantor in set theory—specifically, his discovery of the different sizes of infinity. Cantor showed that infinite sets could be subdivided into those whose elements could be counted in the sense that their elements could be put into one to one correspondence with the natural numbers and those whose elements could not in this sense be counted. The set of integers is countable, as, surprising as it may seem, is the set of rational numbers. The set of real numbers, however, along with the set of complex numbers and the set of all subsets of the natural numbers are all uncountably infinite. Cantor called the size of an infinite set its cardinality [See the entry on the early development of set theory].

The Löwenheim-Skolem Theorem states that if a a set of FOL sentences has an infinite model, it has a model whose domain is countably infinite. The Upward Löwenheim-Skolem Theorem states that if a countable set of FOL sentences has an infinite model of some cardinality \(\kappa\) then it has a model of every infinite cardinality [See the entry Skolem’s paradox for the history of the theorems and the philosophical issues concerning them].

Now realists believe that even a rationally optimal or ‘ideal’ theory of the world could be mistaken. Putnam essays to prove that this belief is incoherent. But why can’t an ideal theory be false? To admit that this is possible is to admit that there is a gap between what is true and what is ideally warranted by our best theory, something no anti-realist can afford. But an argument is needed to show this is not possible. Anti-realists have one in the Model-Theoretic Argument. It proceeds thus:

Suppose we had an ideal theory which passed every observational and
theoretical test we could conceive of. Assume this theory could be
formalized in first-order logic. Assume also that the world is
infinite in size and that our formalized ideal theory \(T\) says
it is. Assume, finally, \(T\) is consistent. Then given these
assumptions, Putnam argues, we can show that \(T\) is also
true:

Firstly, as \(T\) is syntactically consistent, by the
Completeness Theorem for first-order logic, \(T\) will have a
model. Then by the Upward Löwenheim-Skolem Theorem, there exists
a model elementarily equivalent to the model generated by the
Completeness Theorem that is of the same size as the world (since by the Upward
Löwenheim-Skolem Theorem \(T\) will have models
of every infinite size). Call this model \(M\).

Nothing in the construction of \(M\) guarantees that the objects
in its domain are objects in the real world. To the contrary, the
domain of \(M\) may be comprised wholly of real numbers for
example. So to obtain, as required, a model whose domain consists of
objects in the world, use Löwenheim-Skolem Theorem once
more to project the model \(M\) onto the world by generating
from \(M\) a new model \(W\) whose domain consists of the
objects in the world and which assigns to all the predicates
of \(T\) subclasses of its domain and relations defined on that
domain.

We now have a correspondence between the expressions of the
language \(L\) in which \(T\) is expressed and (sets of)
objects in the world just as the realist requires. \(T\) will
then be true if ‘true’ just means
‘true-in-\(W\)’.

If \(T\) is not guaranteed true by this procedure it can only be
because \(W\) is not the intended model. Yet all our
observation sentences come out true according to \(W\) and the
theoretical constraints must be satisfied because T’s
theses all come out true in \(W\) also. So the realist owes us an
explanation of what constraints a model has to satisfy for it to be
‘intended’ over and above its satisfying every
observational and theoretical constraint we can conceive of.

Suppose on the other hand that the realist is able to somehow specify
the intended model. Call this intended model \(W''\). Then nothing the
realist can do can possibly distinguish \(W''\) from a permuted
variant \(W^*\) which can be specified following Putnam 1994b,
356–357:

We define properties of being a cat* and being a
mat* such that:

In the actual world cherries are cats* and trees are
mats*.

In every possible world the two sentences “A cat is on a
mat” and “A cat* is on a mat*”
have precisely the same truth value.

Instead of considering two sentences “A cat is on a mat”
and “A cat* is on a mat*” now
consider only the one “A cat is on a mat”, allowing its
interpretation to change by first adopting the standard interpretation
for it and then adopting the non-standard interpretation in which the
set of cats* are assigned to ’cat’ in every
possible world and the set of mats* are assigned to
’mat’ in every possible world. The result will be the
truth-value of “A cat is on a mat” will not change and
will be exactly the same as before in every possible world. Similar
non-standard reference assignments could be constructed for all the
predicates of a language. [See Putnam 1985, 1994b.]

We now turn to some realist responses to these challenges. The
Manifestation and Language Acquisition arguments allege there is
nothing in an agent’s cognitive or linguistic behaviour that
could provide evidence that they had grasped what it is for a sentence
to be true in the realist’s sense of ‘true’. How can
you manifest a grasp of a notion which can apply or fail to apply
without you being able to tell which? How could you ever learn to use
such a concept?

One possible realist response is that the concept of truth is actually
very simple, and it is spurious to demand that one always be able to
determine whether a concept applies. As to the first part, it is often
argued that all there is to the notion of truth is what is given by
the formula “‘\(p\)’ is true if and only
if \(p\)”. The function of the truth-predicate is to
disquote sentences in the sense of undoing the effects of
quotation—thus all that one is saying in calling the sentence
“Yeti are vicious” true is that Yeti are
vicious.

It is not clear that this response really addresses the
anti-realist’s worry, however. It may well be that there is a
simple algorithm for learning the meaning of ‘true’ and
that, consequently, there is no special difficulty in learning to
apply the concept. But that by itself does not tell us whether the
predicate ‘true’ applies to cases where we cannot
ascertain that it does. All the algorithm tells us, in effect, is that
if it is legitimate to assert \(p\) it is legitimate to assert
that ‘\(p\)’ is true. So are we entitled to assert
‘either Socrates did or did not sneeze in his sleep the night
before he took the hemlock’ or are we not? Presumably that will
depend on what we mean by the sentence, whether we mean to be
adverting to two states of affairs neither of which we have any
prospect of ever confirming.

Anti-realists follow verificationists in rejecting the intelligibility
of such states of affairs and tend to base their rules for assertion
on intuitionistic logic, which rejects the universal applicability of
the Law of Bivalence (the principle that every statement is either
true or false). This law is a foundational semantic principle for
classical logic.

A more direct realist response to the Manifestation challenge points
to the prevalence in our linguistic practices of realist-inspired
beliefs to which we give expression in what we say and do. We assert
things like “either there were an odd or an even number of
dinosaurs on this planet independently of what anyone believes”
and all our actions and other assertions confirm that we really do
believe this. Furthermore, the overwhelming acceptance of classical
logic by mathematicians and scientists and their rejection of
intuitionistic logic for the purposes of mainstream science provides
very good evidence for the coherence and usefulness of a realist
understanding of truth.

Anti-realists reject this reply. They argue that all we make manifest
by asserting things like “either there were an odd or an even
number of dinosaurs on this planet independently of what anyone
believes” is our pervasive misunderstanding of the notion of
truth. They apply the same diagnosis to the realist’s belief in
the mind-independence of entities in the world and to counterfactuals
which express this belief. We overgeneralize the notion of truth,
believing that it applies in cases where it does not, they
contend.

An apparent consequence of their view is that reality is indeterminate
in surprising ways—we have no grounds for asserting that
Socrates did sneeze in his sleep the night before he took the hemlock
and no grounds for asserting that he did not and no prospect of ever
finding out which. Does this mean that for anti-realists the world
contains no such fact as the fact that Socrates did one or the other
of these two things? Not necessarily. For anti-realists who subscribe
to intuitionistic principles of reasoning, the most that can be said
is that there is no present warrant to assert that Socrates either did
or did not sneeze in his sleep the night before he took the
hemlock.

Perhaps anti-realists are right about all this. But if so, they need
to explain how a practice based on a pervasive illusion can be as
successful as modern science. Anti-realists perturbed by the
manifestability of realist truth are revisionists about parts of our
linguistic practice, and the consequence of this revisionist stance is
that mathematics and science require extensive and non-trivial
revision.

Much could be and has been said by anti-realists in response to this
point. Standing back from the debate between the two sides is not
always easy but at least this point should be made. Nothing said so
far solves the Representation Problem, the problem of how our mental
symbols get to target mind-independent entities in the first place,
let alone the right ones. Some natural mechanism for generating the
right links must be at work for it cannot just be a primitive
inexplicable fact that ‘the Big Bang’ refers to the Big
Bang. If this problem could be solved, the Manifestation and
Acquisition challenges would, presumably, be answered. It would then
be the burden of the other pragmatist-inspired anti-realist challenges
to show that the realist cannot solve the Representation Problem.

The challenge to realism posed by language acquisition is to explain
how a child could come to know the meanings of certain sentences
within his/her language: the ones which the realist contends have
undetectable truth-makers associated with them. How could the child
learn the meanings of such sentences if these meanings are determined
by states of affairs not even competent speakers can detect?

How should realists respond to this challenge? They should question
the publicity of meaning principle as it applies to language learning
and they should question this principle on empirical as well as
conceptual grounds. That the meaning of a word is in some sense
determined by its use in a given language is little more than a
platitude. That the meaning of a word is exhaustively manifest in its
use as an instrument of communication is not.

The evidence from developmental psychology indicates some meaning is
pre-linguistic and that some pre-linguistic meaning or conceptual
content does indeed relate to situations that are not detectable by
the child. For example, psychologists have discovered systems of core
knowledge activated in infancy that govern the representation
of, inter alia concrete objects and human agents [see Spelke
2003; Spelke and Kinzler 2007]. An interesting finding from
preferential gaze experiments suggests 4 month old infants represent
occluded objects as continuing behind their barriers.

Even more surprisingly, 2 day old chicks exposed to occluded objects
for the first time do so as well [Spelke 2003]! Chicks who in their
first day of life had imprinted on a centre-occluded object were
placed in an unfamiliar cage on their second day and presented with a
choice between two versions of the object placed at opposite ends of
the cage. In one version, the visible ends were connected; in the
other these ends were separated by a visible gap matching the occluder
they’d seen the day before. “Chicks selectively approached
the connected object, providing evidence that they, like human
infants, had perceived the imprinted object to continue behind its
occluder” [Spelke 2003, p.283].

So there is evidence that ‘verification-transcendent’
conceptual content might be laid down in the earliest stages of
cognitive development.

Recent studies of conflict detection in reasoning suggest the
anti-realist’s restriction of the basis for ascriptions of
meaning to an overtly behavioural one is unwarranted. It is
well-attested that subjects unfamiliar with logic evince belief-bias
when they reason, accepting conclusions as following from premises
only when those conclusions accord with their background beliefs. On
the basis of their inferential dispositions and their think-out-loud
protocols, most psychologists had concluded that these subjects are
simply unaware of the simple logical rules they appear to flout in the
problematic ‘conflict’ cases.

This is not necessarily so, however. In recent years psychologists
using more subtle measures to detect recognition of logical rules have
found that even the poorest reasoners evince wide range of tacit
behavioural and neurological symptoms when they deny the conclusions
of valid conflict arguments: (i) lower confidence levels (ii) higher
arousal (iii) more frequent saccades to the conflict conclusion and
premises (iv) impaired memory access to cued beliefs that conflict
with normative logical rules (v) heightened activity in an area of the
brain, the anterior cingulate cortex, known to be associated with
conflict detection [See for instance De Neys & Glumicic 2008,
Franssens & De Neys 2009].

The Brains-in-a-Vat argument purports to show that, given semantic
externalism, realism is incoherent on the grounds that it is both
committed to the genuine possibility of our being brains in a vat and
yet entails something inconsistent with this: namely, that were we to
be so envatted we could not possibly have the thought that we
were!

Realists have three obvious responses.

Deny realism entails that we could be brains in a vat.

Deny semantic externalism.

Deny there is any inconsistency between our being brains in a vat
and our inability to think that we were brains in a vat were we to be
so.

As for (i), naturalistic realists do question the coherence of the
idea of our being brains in a vat. For them there is no external
vantage point from which one can assess our best overall theory and
yet the skeptic’s hypothesis feigns to occupy just such a
vantage point. How so? By using terms which derive their meaning from
successful theory to pose a problem which, if intelligible, would rob
those very terms of meaning. In a similar vein some naturalistic
realists have claimed that the mad scientists face an insoluble
problem of combinatorial explosion the moment they give you any
significant exploratory and volitional powers in the virtual world in
which you are imprisoned.

As to the latter, it may be that the clever alien scientists have
generated a convincing illusion of significant exploratory and
volitional powers in the mind of the poor envatted brain. Whether the
skeptic’s prospect is intelligible only at the cost of robbing
the very terms in which it is framed of meaning is much more difficult
to assess, however.

What of option (ii)—denying semantic externalism? Is this really
a live prospect for realists? The answer is
“Yes”. Semantic externalism no longer commands the
consensus amongst realists that it did when Putnam formulated his
Brains-in-a-Vat argument—realists are today divided over the
question of externalism. David Lewis, a prominent realist, rejected
externalism in favour of a sophisticated semantic internalism based on
a ‘Two-Dimensional’ analysis of modality. Frank Jackson
[Jackson, F. 2000] contributed to the development of this internalist
2D semantics and used it to formulate a version of materialism
grounded on conceptual analysis that provides a useful and persuasive
model of a naturalistic realist’s metaphysics.

Other realists reject externalism because they think that the
Representation Problem is just a pseudo-problem. When we say things
like “‘cat’ refers to cats” or
“‘quark’ refers to quarks” we are simply
registering our dispositions to call everything we consider
sufficiently cat-like/quark-like,
‘cat’/’quark’.

According to these semantic deflationists, it is just a confusion to
ask how the link was set up between our use of the term ‘the Big
Bang’ and the event of that name which occurred some fourteen
billion years ago. Some naturalistic story can, presumably, be told
about how creatures like us developed the linguistic dispositions we
did, in the telling of which it will emerge how we come to assert
things like “the entropy of the Big Bang was very
low”.

But it is a moot question whether semantic deflationism really
dissolves the Representation Problem or merely fails to face up to
it. However the story about the origins of our linguistic dispositions
is told, it had better be that our utterances of “the entropy of
the Big Bang was very low” somehow end up evincing just the
right sort of differential sensitivity to the Big Bang’s having
low entropy. For if all there is to the story are our linguistic
dispositions and the conditions to which they are presently attuned,
the case has effectively been ceded to the anti-realist who denies it
is possible to set up a correlation between our utterances or thoughts
and the mind-independent states of affairs which, according to the
metaphysical realist, uniquely make them true.

The most effective realist rejoinder is (iii). We shall return to this
response after we have reviewed Putnam’s Brains-in-a-Vat
Argument, BIVA.

How does Putnam prove we can know we are not brains in a vat? To
understand Putnam’s argument, we need to first recall the
‘Twin-Earth’ considerations used to support Semantic
Externalism: on Twin-Earth things are exactly as they are here on
Earth except for one difference—whereas for Earthly humans
water has the chemical composition H2O, for our
döppelgangers on Twin-Earth, twumans, water is instead composed
of some substance unknown to us on Earth, XYZ. Now when you and your
twuman counterpart say (or think) “’Water’ refers to
water” both of you utter (or think) truths. But which truth you
both think or utter differs. For humans “’Water’
refers to water” expresses the truth that the term
‘water’ in English refers to that substance whose chemical
composition is H2O. For our twuman Twin-Earth counterparts,
however, their sentence “’Water’ refers to
water” expresses the truth that their term ‘water’
in Twenglish refers to that substance whose chemical composition is
XYZ.

With these points about Externalism in mind, consider Putnam’s
BIVA [we follow Anthony Brueckner’s formulation here: see the
entry skepticism and content
externalism]. Let us call whatever it is that an envatted
brain’s symbol ‘tree’ refers to, if it refers at
all, \(v\)-trees. Then the BIVA is:

(1)
If I’m a BIV then it is not the case that if my word
‘tree’ refers it refers to trees.

(2)
If my word ‘tree’ refers it refers to trees.

(3)
So, I am not a BIV.

Now (1) is correct: if I am a brain-in-a-vat then my symbol
‘tree’ cannot refer to trees since there aren’t any
trees in the vat-world—a BIV’s ‘tree’ symbol
refers to \(v\)-trees, not trees. But what reason do we have
to believe (2)?

Just as we can do, our twuman döppelgangers on Twin-Earth can
justifiably declare “If my word ‘water’ refers it
refers to water.” But despite the fact that the twumans’
language Twenglish is a homophonic duplicate of English,
‘water’ on Twin-Earth refers to twater, XYZ, not
water, H2O.

Instead of (2) we really need two premises incorporating distinct
hypotheses about the language I am speaking:

(2E)
If the language I am speaking is English and my word
‘tree’ refers, it refers to trees.

(2V)
If the language I am speaking is Venglish (Vattish English) and
my word ‘tree’ refers, it refers to trees.

To preserve the logical form of the original argument, (1) must also
be modified to accept each hypothesis about the language I am speaking
so that it now bifurcates into:

(1E)
(1E) If I’m a BIV then it is not the case that if the language I
am speaking is English and my word ‘tree’ refers it refers
to trees.

(1V)
If I’m a BIV then it is not the case that if the language I
am speaking is Venglish and my word ‘tree’ refers it
refers to trees.

The problem with Putnam’s BIVA is that while (1V) is true, (2V)
is false whereas although (2E) is true, (1E) is false. On either
hypothesis about the language I am speaking, it appears that there is
no sound argument to the conclusion that I am not a
brain-in-a-vat.

Suppose the BIVA is unsound. What would this show? Even if the BIVA
fails to achieve its goal, Putnam’s challenge to the realist
remains unanswered. This was to show how realism could be coherent if
it is committed both to:

(I)
The real possibility that we are brains-in-a-vat

and to the consequence that:

(II)
Were we to be BIVs we could not have the thought that we were.

In fact, there is no logical incoherence in accepting both (I)
and (II)—as the figure below illustrates. There is thus no
logical incoherence in believing both that it is possible that one is
a BIV and that if one is a BIV one could never come to know this.

Figure. If we are not in fact
brains-in-a-vat (so that the hypothesis \(v\) that we are brains in a
vat is false, \(\neg v\) is true at \(w^*\), the actual world) we can
nonetheless entertain \((E)\), the hypothesis that we are (so that
\(Ev\) is true at \(w^*\)), recognizing as we do so that were we to
inhabit a world such as \(w'\) in which we are brains-in-a-vat (\(v\)
holds at \(w'\), we would lack the semantic resources to articulate
thoughts reflecting our own envatted state so that we could not
so much as entertain the thought that we were brains-in-a-vat (\(\neg
Ev\) holds not only at \(w'\) but at all worlds accessible from \(w'\)
such as \(w''\), \(w'''\), etc).

Nick Bostrom has recently argued it is quite likely that we humans are
actually virtual humans: computer simulations of flesh and
blood creatures. Bostrom reasons that if our mental lives can be
simulated it is in fact extremely likely that our distant descendants
(more intelligent or at least more technologically advanced
‘post-human’ successors) will eventually create such a
simulation in which case it is more likely that we are the unwitting
denizens of a simulated world than the flesh and blood inhabitants of
the real world we take ourselves to be. At least this will be so
unless the chances that creatures of our intelligence are doomed to
become extinct before reaching the technological sophistication to
create simulations are overwhelmingly large or else almost no such
technologically capable civilizations have any interest in simulating
minds like ours in the first place [Bostrom, N., 2003].

Bostrom’s argument makes it look unlikely that we can know a
priori that we are not brains-in-a-vat, when BIVs are
understood to be virtual humans in a simulation. If this is correct,
Putnam’s attempt to prove we are not BIVs must be
flawed. Whether the Simulation Argument poses further problems for
realism is a moot point.

In this section I shall outline the anti-realist’s idea of
conceptual relativity and indicate some ways realists might wish to
contest that notion. I shall not try to distinguish between conceptual
relativism and conceptual pluralism. Conceptual relativism looks
highly counter-intuitive to realists since it seems to make the
existence of all things relative to the classificatory skills of
minds. Whilst it may be quite plausible to think that moral values or
perhaps even colours might disappear with the extinction of sentient
life on Earth, it is not at all plausible to think that trees, rocks
and microbes would follow in their train. If that is what it commits
us to, then the idea of conceptual relativity looks highly
suspect.

This is not how anti-realists understand conceptual relativity,
however. As they see things, we accept a theory which licenses us to
assert “ Electrons exist ” and also licenses us to assert
“if humans were to disappear from this planet, electrons need
not follow in their train” since the theory assures us that the
existence of electrons in no way causally depends on the existence of
humans. For the anti-realist our well-founded practices of assertion
ground at one and the same time our conception of the world and our
conception of humanity’s place within it.

Realists might still worry that whether there are to be any electrons
in the anti-realist’s ontology apparently depends upon the
conceptual schemes humans happen to chance upon. The relativity of
existence to conceptual scheme is, in this respect, quite unlike the
relativity of simultaneity to frame of reference.

Still, we have actual instances of conceptual schemes which explain
the same phenomena equally well yet which realists must adjudge
logically incompatible anti-realists maintain. The earlier example of
competing theories of space-time was a case in point. On one theory,
space-time consists of unextended spatiotemporal points and regions of
space-time are sets of these points. According to the second theory,
space-time consists of extended spatiotemporal regions and points are
logical constructions—convergent sets of regions.

Anti-realists regard two theories as descriptively equivalent if each
theory can be interpreted in the other and both theories explain the
same phenomena. Is there nothing more to the notion of descriptive
equivalence than this? Realists might not accept that there
isn’t.

At the stroke of midnight Cinderella’s carriage changes into a
pumpkin—it is a carriage up to midnight, a pumpkin
thereafter. According to the region-based theory which takes temporal
intervals as its primitives, that’s all there is to it. But if
there are temporal points, instants, there is a further fact left
undecided by this story—viz, at the moment of midnight is the
carriage still a carriage or is it a pumpkin?

So does the region-based theory fail to recognize certain facts or are
these putative facts merely artefacts of the punctate theory’s
descriptive resources, reflecting nothing in reality? We cannot
declare the two theories descriptively equivalent until we resolve
this question at least.

In general, then, realists either dismiss cases of apparent logical
incompatibility between two descriptively equivalent rival theories as
merely apparent or question the descriptive equivalence of the two
theories.

The conceptual relativity we have been discussing has its roots in
Carnap’s views about linguistic frameworks. As we saw in section
2, Carnap rejected the idea that we could answer existence questions
in any absolute sense. If we ask, as we did before, whether there are
irrational numbers \(a, b\) such
that \(a^b\) is rational, Carnap requires we
first specify a framework before tendering any reply. If we choose
classical mathematics the answer is “Yes” but if we choose
intuitionistic mathematics, the answer is “There is no warrant
for asserting such \(a\) and \(b\) exist.” So
according to Carnap whilst the claim that irrational
numbers \(a, b\) such
that \(a^b\) is rational exist-in-CM
is perfectly true, the claim that such \(a, b\)
exist simpliciter is meaningless.

We saw in section 2, though, that the questions dividing realists from
anti-realists appear to survive indexation to frameworks. Revisionary
intuitionists who object to non-constructive existence proofs in
mathematics are not just expressing a preference for
constructive methods: they find the notion of non-constructive
existence unintelligible not just unappealing:

So consider this case: Ernie looks into his bag and sees there are 3
coins and nothing else, so he announces “There are exactly 3
objects in my bag.” Max looks into Ernie’s bag and shakes
his head “No Ernie there are 7 objects in your bag!” he
corrects him. The Carnapian pluralist feels she can defuse the
conflict and accommodate both points of view by maintaining that
whilst 3 objects exist-in-\(E\) (where \(E\) is
Ernie’s everyday framework), 7 objects exist-in-\(M\)
(with \(M\) Max’s mereological framework). But even if Max
can endorse both of these claims (since the mereological objects
include Ernie’s 3 marbles), it is not at all certain Ernie can
do so. If Ernie is unpersuaded that mereological fusions of objects
are themselves objects, then Max’s putative truthmaker for his
framework-relative existence claim “7 objects
exist-in-\(M\)” will be unconvincing to him.

For this case, there seems little reason to accept the
pluralist’s description that whilst 3 objects
exist-in-\(E\), 7 objects exist-in-\(M\) and some good
reasons not to. By ‘object’ Ernie means ordinary object,
by ‘object’ Max means mereological object. Nothing deeper
than that is required to explain their disagreement. What is
relativized in the first instance is not existence or truth but
meaning. To the non-relativist, it looks as if pluralists have simply
mistaken a plurality of meanings for a plurality of modes of
being.

This is not to say that all there ever is to such disputes is a
misunderstanding about the meanings of words. There is still the
substantive question of which of two theories conceived as rivals is
true. Thus in the dispute between classical and intuitionist logicians
the attempt to import distinctive intuitionistic connectives into
classical languages containing classical connectives results in the
‘intuitionistic’ connectives obeying classical logical
laws.

The point is rather that whether there are mereological (or
ordinary) objects ought not to be prejudged by stipulating they exist
within some framework nor can it be resolved satisfactorily by this
means. By way of comparison, consider the question of whether
space-time is continuous or discrete. This looks like a substantive
empirical question. If String Theory which says space-time is
continuous and Quantum Loop Gravity Theory which says it is discrete
were to prove equally good ‘final’ theories of space-time
as judged by all the evidence, we would naturally take this to
demonstrate that the matter of whether space-time was a continuum had
turned out to be empirically undecidable, not that space-time had
turned out in one sense to be continuous (continuousST) but
in another to be discrete (discreteQLGT)—if we can
know this type of thing at all, this is something we can know a
priori without heeding any evidence. Rightly or wrongly, we
would take the question “Is Space-Time continuous or
discrete?” to remain unsolved in the envisaged circumstances,
rather than equally well though divergently solved. We might then go
on to cite this case as a good example of underdetermination of theory
by evidence.

Moreover, the space-time and mereological examples bring to light
another problem for pluralism. If there is no framework-neutral
metalanguage neo-Carnapians can deploy to state their purported
framework-relative ontological truths, how can there be a fact to the
matter as to what things exist in a framework? Realists might then
wonder whether the pluralist’s position doesn’t threaten
to become ineffable.

If metaphysical realism is to be tenable, it must be possible for even
the best theories to be mistaken. Or so metaphysical realists have
thought. Whence, such realists reject the Model-Theoretic Argument MTA
which purports to show that this is not possible. Here is an informal
sketch of the MTA due to van Fraassen [1997]:

Let \(T\) be a theory that contains all the sentences we insist
are true, and that has all other qualities we desire in an ideal
theory. Suppose moreover that there are infinitely many things, and
that \(T\) says so. Then there exist functions (interpretations)
which assign to each term in \(T\)’s vocabulary an
extension, and which satisfy \(T\). So we conclude, to quote
Putnam, “\(T\) comes out true, true of the world, provided
we just interpret ‘true’ as TRUE(SAT)”.

Here ‘TRUE(SAT)’ means “true relative to a mapping
of the terms of the language of \(T\) onto (sets of) items in the
world”.

But why should we interpret ‘true’ as TRUE(SAT)? Because
truth is truth in an intended model and, Putnam argues, amongst all
the models of \(T\) that make all its theses come out true there
is guaranteed to be at least one that passes all conceivable
constraints we can reasonably impose on a model in order for it to be
an intended model of \(T\).

Realists have responded to the argument by rejecting the claim that a
model \(M\) of the hypothetical ideal theory \(T\) passes
every theoretical constraint simply because all of the theory’s
theses come out true in it. For there is no guarantee, they claim,
that terms stand in the right relation of reference to the objects to
which \(M\) links them. To be sure, if we impose another
theoretical constraint, say:

Right Reference Constraint (RRC): Term \(t\) refers to
object \(x\) if and only if \(Rtx\) where \(R\) is the right
relation of reference,

then \(M\) (or some model based on it) can interpret this RRC
constraint in such a way as to make it come out true.

But there is a difference between a model’s making some
description of a constraint come out true and its actually conforming
to that constraint, metaphysical realists insist [Devitt 1983, 1991;
Lewis 1983, 1984].

For their part, anti-realists have taken the metaphysical
realist’s insistence on a Right Reference Constraint to be
‘just more theory’—what it is for a model to
conform to a constraint is for us to be justified in asserting that it
does. Unfortunately, this has led to something of a
stand-off. Metaphysical realists think that anti-realists are refusing
to acknowledge a clear and important distinction. Anti-realists think
realists are simply falling back on dogmatism at a crucial point in
the argument.

On the face of it, the Permutation Argument presents a genuine
challenge to any realist who believes in determinate reference. But it
does not refute metaphysical realism unless such realism is committed
to determinate reference in the first place and it is not at all
obvious that this is so.

Realist responses to this argument vary widely. At one extreme are the
‘determinatists’, those who believe that Nature has set up
significant, determinate referential connections between our mental
symbols and items in the world. They contend that all the argument
shows is that the distribution of truth-values across possible worlds
is not sufficient to determine reference.

At another extreme are ‘indeterminatists’, realists who
concede the conclusion, agreeing that it demonstrates that word-world
reference is massively indeterminate or ‘inscrutable’.

Some infer from this that reference could not possibly consist in
correspondences between mental symbols and objects in the world. For
them all that makes ‘elephant’ refer to elephants is that
our language contains the word ‘elephant’. This is
Deflationism about reference.

In between these two extremes are those prepared to concede the
argument establishes the real possibility of a significant and
surprising indeterminacy in the reference of our mental symbols but
who take it to be an open question whether other constraints can be
found which pare down the range of reference assignments to just the
intuitively acceptable ones. On this view ‘elephant’ may
partially refer to elephants according to one acceptable reference
assignment and may partially refer to elephant-stages or undetached
elephant parts according to other such assignments, but not refer,
even partially, to quolls or quarks. In this spirit, Hartry Field
[1998] has argued that an objective referential indeterminacy he calls
‘correlative indeterminacy’ could exist quite undetected
in linguistic practices such as ours that assume determinacy of
reference for terms:

If ‘entropy’ partially refers to \(E_1\)
and \(E_2\), then we can say that relative to an
assignment of \(E_1\) to ‘entropy’,
‘refers’ refers to a relation that holds between
‘entropy’ and only one thing, viz. \(E_1\);
and analogously for \(E_2\). In this way we can get the
result that even if ‘entropy’ partially refers to many
things (and hence does not determinately refer to anything), still the
sentence “‘Entropy’ refers to entropy and nothing
else” comes out true. (Indeed, determinately true: true on every
acceptable combination of the partial referents of
‘entropy’ and ‘refers’). The advocate of
indeterminacy can still ‘speak with the
vulgar’. [loc. cit., p. 254]

The simplest and most direct response to the MTA questions its
validity—since all versions of the MTA challenge the realist to
say why terms are not related to their right referents in the
alternative models Putnam constructs, metaphysical realists have been
quick to respond. Thus Devitt and Lewis claim that Putnam’s
alternative model \(M\) has not been shown to
satisfy every theoretical constraint merely by
making some description of each theoretical constraint true.

Skolem’s Paradox in set theory seems to present a striking
illustration of Lewis’s distinction. The Löwenheim-Skolem
Theorem states that every consistent, countable set of first-order
formulae has a denumerable model, in fact a model in the set of
integers \(\mathbb{Z}\). Now in ZF one can prove the existence of sets with a
non-denumerable number of elements such as the set \(\mathbb{R}\) of real
numbers. Yet the ZF axioms comprise a consistent, countable set of
first-order formulae and thus by the Löwenheim-Skolem Theorem has
a model in \(\mathbb{Z}\). So ZF’s theorem \(\phi\) stating that
\(\mathbb{R}\) is non-denumerable will come out true in a denumerable model
\(\mu\) of ZF!

How can this be? One explanation is that \(\mu\) makes \(\phi\)
true only at the cost of re-interpreting the term
‘non-denumerable’ so that it no longer
means non-denumerable. Thus \(\mu\) is not the intended
model \(M^*\) of ZF. It looks as if the metaphysical
realist has a clear illustration of Lewis’s distinction at hand
in set theory.

Unfortunately for the realist, this is not the only explanation. In
fact, Putnam used this very example in an early formulation of the
MTA! Just because there are different models that
satisfy \(\phi\) in some of which \(\mathbb{R}\) is non-denumerable but
in others of which (such as \(\mu\)) \(\mathbb{R}\) is denumerable, Putnam argued, it is impossible to pin down the intended interpretation of
‘set’ via first-order axioms. Moreover, well before Putnam, Skolem and his followers had taken the moral of Skolem’s
Paradox to be that set-theoretic notions are indeterminate [For further discussion, see the entry on Skolem’s paradox].

Now the thesis the metaphysical realist has to establish is that an
ideal theory could be false. If truth for our imagined ideal theory
\(T\) is truth on its intended model(s) \(M^*\) this amounts to the
possibility that some thesis of \(T\) come out false in \(M^*\) even
if there are other models wherein that thesis along with every other
thesis of \(T\) comes out true. This brings us to the Right Reference
Constraint (RRC) once more and the question of what it is for \(T\) to
satisfy (RRC). Discussing this issue in connection with set theory
Timothy Bays [2001] writes:

When a philosopher claims that the intended models of set theory
should be transitive, she is describing the structures which
are to count as models for her axioms; she isn’t just adding new
sentences to be interpreted at Putnam’s favorite
models. Similarly, when she claims that intended models should satisfy
second-order ZFC, she is explaining which semantics (and,
more specifically, which satisfaction relation) her axioms should be
interpreted under; she isn’t just adding new axioms to be
interpreted under a (first-order) semantics of Putnam’s
choosing.

Putnam [1985] responded to this point by charging the realist with
question-begging: simply assuming that terms such as
‘satisfaction’ or ‘correspondence’ refer to
those relations to which the realist wishes them to refer. But, as
Bays points out, no questions are begged if realists assume for the
purposes of argument that semantic terms such as these refer to the
intended relations. It is up to the anti-realist to show that this
assumption is flawed. Nonetheless, merely invoking an intuitive
distinction does not show the distinction marks a genuine
difference. So if realism is to be sustained, there had better be some
more convincing or, at any rate, less contentious examples than
Skolem’s Paradox to illustrate Lewis’s alleged difference
between a model M’s satisfying a constraint and
M’s merely making a description of the constraint true. Are
there?

Michael Resnick thinks so [Resnick 1987]. Putnam maintained
that \(M\), the model he constructs of the ideal
theory \(T\), is an intended model because it passes every
operational and theoretical constraint we could reasonably impose. It
passes every theoretical constraint, he argues, simply because it
makes every thesis of \(T\) true. But unless the Reflection
Principle (RP) below holds, Resnick argues, this inference is
just a non-sequitur:

(RP)
To any condition \(f\) that a model of a theory
satisfies, there corresponds a condition \(C\) expressible in the
theory that that theory satisfies.

However, this principle is false. The simplest counterexample to it,
Resnick points out, is Tarskian truth. Suppose we impose
on \(T\)’s model \(M\) a
condition \(f^*\) that \(M\) makes all
of \(T\)’s theses come out true. Then, unless \(T\) is
either inconsistent or too weak to express elementary arithmetic no
truth predicate will be definable in \(T\). Whence there will be
no condition \(C\) expressible in \(T\) corresponding to
this condition \(f^*\) on \(T\)’s
model(s) \(M\).

Resnick concludes (ibid):

Any true interpretation of \(T\) whatsoever—even one which
does not satisfy \(C\)—will make true every thesis
of \(T\), including T’s assertion that \(C\)
is satisfied. Which suffices to block the ‘just more
theory’ gambit.

If this is right, the metaphysical realist can indeed resist what
Lewis calls “Putnam’s incredible thesis” that an
ideal theory \(T\) has to be true. More recently, there
have been some sophisticated anti-realist attempts to buttress the
Model-Theoretic Argument against Lewis-styled criticisms [Taylor 2006;
Button 2013]. Whether these newer formulations of the MTA succeed in
doing so is an open question.

We have considered a number of semantic challenges to realism, the
thesis that the objects and properties that the world contains exist
independently of our conception or perception of them. These
challenges have come from two camps: (1) neo-verificationists led by
Dummett who assimilate belief in mind-independent world to a belief in
a verification-transcendent conception of truth which they profess to
find unintelligible, and (2) pragmatists led by Putnam who also
question the intelligibility of the realist’s mind-independent
world but for reasons independent of any commitment to
verificationism.

On all fronts, debate between realists and their anti-realist
opponents is still very much open. If realists could provide a
plausible theory about how correspondences between mental symbols and
the items in the world to which they refer might be set up, many of
these challenges could be met. Alternatively, if they could explain
how, consistently with our knowledge of a mind-independent world, no
such correspondences are required to begin with, many of the
anti-realist objections would fall away as irrelevant. In the absence
of such explanations it is still entirely reasonable for realists to
believe that the correspondences are in place, however, and there can,
indeed, be very good evidence for believing this. Ignorance of
Nature’s reference-fixing mechanism is no reason for denying it
exists.

Acknowledgments

Thanks to Jesse Alama and a subject editor for the Stanford
Encyclopedia of Philosophy for their helpful criticisms and
corrections. Special thanks to Marinus Ferreira for many useful
suggestions and for help in writing and editing this entry.